Solving initial value problems for stiff or non-stiff systems of first-order ordinary differential equations (ODEs).
The R function lsoda provides an interface to the FORTRAN ODE
solver of the same name, written by Linda R. Petzold and Alan
C. Hindmarsh.
The system of ODE's is written as an R function (which may, of
course, use .C, .Fortran,
.Call, etc., to call foreign code) or be defined in
compiled code that has been dynamically loaded. A vector of
parameters is passed to the ODEs, so the solver may be used as part of
a modeling package for ODEs, or for parameter estimation using any
appropriate modeling tool for non-linear models in R such as
optim, nls, nlm or
nlme
lsoda differs from the other integrators (except lsodar)
in that it switches automatically between stiff and nonstiff methods.
This means that the user does not have to determine whether the
problem is stiff or not, and the solver will automatically choose the
appropriate method. It always starts with the nonstiff method.
lsoda(y, times, func, parms, rtol = 1e-6, atol = 1e-6,
jacfunc = NULL, jactype = "fullint", rootfunc = NULL,
verbose = FALSE, nroot = 0, tcrit = NULL,
hmin = 0, hmax = NULL, hini = 0, ynames = TRUE,
maxordn = 12, maxords = 5, bandup = NULL, banddown = NULL,
maxsteps = 5000, dllname = NULL, initfunc = dllname,
initpar = parms, rpar = NULL, ipar = NULL, nout = 0,
outnames = NULL, forcings = NULL, initforc = NULL,
fcontrol = NULL, events = NULL, lags = NULL,...)the initial (state) values for the ODE system. If y
has a name attribute, the names will be used to label the output
matrix.
times at which explicit estimates for y are
desired. The first value in times must be the initial time.
either an R-function that computes the values of the derivatives in the ODE system (the model definition) at time t, or a character string giving the name of a compiled function in a dynamically loaded shared library.
If func is an R-function, it must be defined as:
func <- function(t, y, parms,...). t is the current
time point in the integration, y is the current estimate of
the variables in the ODE system. If the initial values y has
a names attribute, the names will be available inside func.
parms is a vector or list of parameters; ... (optional) are
any other arguments passed to the function.
The return value of func should be a list, whose first
element is a vector containing the derivatives of y with
respect to time, and whose next elements are global values
that are required at each point in times. The derivatives
must be specified in the same order as the state variables y.
If func is a string, then dllname must give the name
of the shared library (without extension) which must be loaded
before lsoda() is called. See package vignette
"compiledCode" for more
details.
vector or list of parameters used in func or
jacfunc.
relative error tolerance, either a scalar or an array as
long as y. See details.
absolute error tolerance, either a scalar or an array as
long as y. See details.
if not NULL, an R function, that computes the
Jacobian of the system of differential equations
\(\partial\dot{y}_i/\partial y_j\), or
a string giving the name of a function or subroutine in
dllname that computes the Jacobian (see vignette
"compiledCode" for more about this option).
In some circumstances, supplying
jacfunc can speed up the computations, if the system is
stiff. The R calling sequence for jacfunc is identical to
that of func.
If the Jacobian is a full matrix, jacfunc should return a
matrix \(\partial\dot{y}/\partial y\), where the ith row contains the derivative of
\(dy_i/dt\) with respect to \(y_j\), or a vector containing the
matrix elements by columns (the way R and FORTRAN store matrices).
If the Jacobian is banded, jacfunc should return a matrix
containing only the nonzero bands of the Jacobian, rotated
row-wise. See first example of lsode.
the structure of the Jacobian, one of "fullint",
"fullusr", "bandusr" or "bandint" - either
full or banded and estimated internally or by user.
if not NULL, an R function that computes the
function whose root has to be estimated or a string giving the name
of a function or subroutine in dllname that computes the root
function. The R calling sequence for rootfunc is identical
to that of func. rootfunc should return a vector with
the function values whose root is sought. When rootfunc is
provided, then lsodar will be called.
if TRUE: full output to the screen, e.g. will
print the diagnostiscs of the integration - see details.
only used if dllname is specified: the number of
constraint functions whose roots are desired during the integration;
if rootfunc is an R-function, the solver estimates the number
of roots.
if not NULL, then lsoda cannot integrate
past tcrit. The FORTRAN routine lsoda overshoots its
targets (times points in the vector times), and interpolates
values for the desired time points. If there is a time beyond which
integration should not proceed (perhaps because of a singularity),
that should be provided in tcrit.
an optional minimum value of the integration stepsize. In
special situations this parameter may speed up computations with the
cost of precision. Don't use hmin if you don't know why!
an optional maximum value of the integration stepsize. If
not specified, hmax is set to the largest difference in
times, to avoid that the simulation possibly ignores
short-term events. If 0, no maximal size is specified.
initial step size to be attempted; if 0, the initial step size is determined by the solver.
logical, if FALSE: names of state variables are not
passed to function func; this may speed up the simulation especially
for large models.
the maximum order to be allowed in case the method is
non-stiff. Should be <= 12. Reduce maxord to save storage space.
the maximum order to be allowed in case the method is stiff. Should be <= 5. Reduce maxord to save storage space.
number of non-zero bands above the diagonal, in case the Jacobian is banded.
number of non-zero bands below the diagonal, in case the Jacobian is banded.
maximal number of steps per output interval taken by the solver.
a string giving the name of the shared library
(without extension) that contains all the compiled function or
subroutine definitions refered to in func and
jacfunc. See package vignette "compiledCode".
if not NULL, the name of the initialisation function
(which initialises values of parameters), as provided in
dllname. See package vignette "compiledCode".
only when dllname is specified and an
initialisation function initfunc is in the dll: the
parameters passed to the initialiser, to initialise the common
blocks (FORTRAN) or global variables (C, C++).
only when dllname is specified: a vector with
double precision values passed to the dll-functions whose names are
specified by func and jacfunc.
only when dllname is specified: a vector with
integer values passed to the dll-functions whose names are specified
by func and jacfunc.
only used if dllname is specified and the model is
defined in compiled code: the number of output variables calculated
in the compiled function func, present in the shared
library. Note: it is not automatically checked whether this is
indeed the number of output variables calculated in the dll - you have
to perform this check in the code. See package vignette
"compiledCode".
only used if dllname is specified and
nout > 0: the names of output variables calculated in the
compiled function func, present in the shared library.
These names will be used to label the output matrix.
only used if dllname is specified: a list with
the forcing function data sets, each present as a two-columned matrix,
with (time,value); interpolation outside the interval
[min(times), max(times)] is done by taking the value at
the closest data extreme.
See forcings or package vignette "compiledCode".
if not NULL, the name of the forcing function
initialisation function, as provided in
dllname. It MUST be present if forcings has been given a
value.
See forcings or package vignette "compiledCode".
A list of control parameters for the forcing functions.
See forcings or vignette compiledCode.
A matrix or data frame that specifies events, i.e. when the value of a state variable is suddenly changed. See events for more information.
additional arguments passed to func and
jacfunc allowing this to be a generic function.
A matrix of class deSolve with up to as many rows as elements
in times and as many columns as elements in y plus the number of "global"
values returned in the next elements of the return from func,
plus and additional column for the time value. There will be a row
for each element in times unless the FORTRAN routine `lsoda'
returns with an unrecoverable error. If y has a names
attribute, it will be used to label the columns of the output value.
All the hard work is done by the FORTRAN subroutine lsoda,
whose documentation should be consulted for details (it is included as
comments in the source file src/opkdmain.f). The implementation
is based on the 12 November 2003 version of lsoda, from Netlib.
lsoda switches automatically between stiff and nonstiff
methods. This means that the user does not have to determine whether
the problem is stiff or not, and the solver will automatically choose
the appropriate method. It always starts with the nonstiff method.
The form of the Jacobian can be specified by jactype which can
take the following values:
a full Jacobian, calculated internally by lsoda, the default,
a full Jacobian, specified by user function jacfunc,
a banded Jacobian, specified by user function jacfunc
the size of the bands specified by bandup and banddown,
banded Jacobian, calculated by lsoda; the size of the bands
specified by bandup and banddown.
If jactype = "fullusr" or "bandusr" then the user must supply a
subroutine jacfunc.
The following description of error control is adapted from the
documentation of the lsoda source code
(input arguments rtol and atol, above):
The input parameters rtol, and atol determine the error
control performed by the solver. The solver will control the vector
e of estimated local errors in y, according to an
inequality of the form max-norm of ( e/ewt ) \(\leq\) 1, where ewt is a vector of positive error weights. The
values of rtol and atol should all be non-negative. The
form of ewt is:
$$\mathbf{rtol} \times \mathrm{abs}(\mathbf{y}) + \mathbf{atol}$$
where multiplication of two vectors is element-by-element.
If the request for precision exceeds the capabilities of the machine,
the FORTRAN subroutine lsoda will return an error code; under some
circumstances, the R function lsoda will attempt a reasonable
reduction of precision in order to get an answer. It will write a
warning if it does so.
The diagnostics of the integration can be printed to screen
by calling diagnostics. If verbose = TRUE,
the diagnostics will written to the screen at the end of the integration.
See vignette("deSolve") for an explanation of each element in the vectors containing the diagnostic properties and how to directly access them.
Models may be defined in compiled C or FORTRAN code, as well as
in an R-function. See package vignette "compiledCode" for details.
More information about models defined in compiled code is in the package vignette ("compiledCode"); information about linking forcing functions to compiled code is in forcings.
Examples in both C and FORTRAN are in the dynload subdirectory
of the deSolve package directory.
Hindmarsh, Alan C. (1983) ODEPACK, A Systematized Collection of ODE Solvers; in p.55--64 of Stepleman, R.W. et al.[ed.] (1983) Scientific Computing, North-Holland, Amsterdam.
Petzold, Linda R. (1983) Automatic Selection of Methods for Solving Stiff and Nonstiff Systems of Ordinary Differential Equations. Siam J. Sci. Stat. Comput. 4, 136--148.
Netlib: http://www.netlib.org
lsode, which can also find a root
lsodes, lsodar, vode,
daspk for other solvers of the Livermore family,
ode for a general interface to most of the ODE solvers,
ode.band for solving models with a banded
Jacobian,
ode.1D for integrating 1-D models,
ode.2D for integrating 2-D models,
ode.3D for integrating 3-D models,
diagnostics to print diagnostic messages.
# NOT RUN {
## =======================================================================
## Example 1:
## A simple resource limited Lotka-Volterra-Model
##
## Note:
## 1. parameter and state variable names made
## accessible via "with" function
## 2. function sigimp accessible through lexical scoping
## (see also ode and rk examples)
## =======================================================================
SPCmod <- function(t, x, parms) {
with(as.list(c(parms, x)), {
import <- sigimp(t)
dS <- import - b*S*P + g*C #substrate
dP <- c*S*P - d*C*P #producer
dC <- e*P*C - f*C #consumer
res <- c(dS, dP, dC)
list(res)
})
}
## Parameters
parms <- c(b = 0.0, c = 0.1, d = 0.1, e = 0.1, f = 0.1, g = 0.0)
## vector of timesteps
times <- seq(0, 100, length = 101)
## external signal with rectangle impulse
signal <- as.data.frame(list(times = times,
import = rep(0,length(times))))
signal$import[signal$times >= 10 & signal$times <= 11] <- 0.2
sigimp <- approxfun(signal$times, signal$import, rule = 2)
## Start values for steady state
y <- xstart <- c(S = 1, P = 1, C = 1)
## Solving
out <- lsoda(xstart, times, SPCmod, parms)
## Plotting
mf <- par("mfrow")
plot(out, main = c("substrate", "producer", "consumer"))
plot(out[,"P"], out[,"C"], type = "l", xlab = "producer", ylab = "consumer")
par(mfrow = mf)
## =======================================================================
## Example 2:
## from lsoda source code
## =======================================================================
## names makes this easier to read, but may slow down execution.
parms <- c(k1 = 0.04, k2 = 1e4, k3 = 3e7)
my.atol <- c(1e-6, 1e-10, 1e-6)
times <- c(0,4 * 10^(-1:10))
lsexamp <- function(t, y, p) {
yd1 <- -p["k1"] * y[1] + p["k2"] * y[2]*y[3]
yd3 <- p["k3"] * y[2]^2
list(c(yd1, -yd1-yd3, yd3), c(massbalance = sum(y)))
}
exampjac <- function(t, y, p) {
matrix(c(-p["k1"], p["k1"], 0,
p["k2"]*y[3],
- p["k2"]*y[3] - 2*p["k3"]*y[2],
2*p["k3"]*y[2],
p["k2"]*y[2], -p["k2"]*y[2], 0
), 3, 3)
}
## measure speed (here and below)
system.time(
out <- lsoda(c(1, 0, 0), times, lsexamp, parms, rtol = 1e-4,
atol = my.atol, hmax = Inf)
)
out
## This is what the authors of lsoda got for the example:
## the output of this program (on a cdc-7600 in single precision)
## is as follows..
##
## at t = 4.0000e-01 y = 9.851712e-01 3.386380e-05 1.479493e-02
## at t = 4.0000e+00 y = 9.055333e-01 2.240655e-05 9.444430e-02
## at t = 4.0000e+01 y = 7.158403e-01 9.186334e-06 2.841505e-01
## at t = 4.0000e+02 y = 4.505250e-01 3.222964e-06 5.494717e-01
## at t = 4.0000e+03 y = 1.831975e-01 8.941774e-07 8.168016e-01
## at t = 4.0000e+04 y = 3.898730e-02 1.621940e-07 9.610125e-01
## at t = 4.0000e+05 y = 4.936363e-03 1.984221e-08 9.950636e-01
## at t = 4.0000e+06 y = 5.161831e-04 2.065786e-09 9.994838e-01
## at t = 4.0000e+07 y = 5.179817e-05 2.072032e-10 9.999482e-01
## at t = 4.0000e+08 y = 5.283401e-06 2.113371e-11 9.999947e-01
## at t = 4.0000e+09 y = 4.659031e-07 1.863613e-12 9.999995e-01
## at t = 4.0000e+10 y = 1.404280e-08 5.617126e-14 1.000000e+00
## Using the analytic Jacobian speeds up execution a little :
system.time(
outJ <- lsoda(c(1, 0, 0), times, lsexamp, parms, rtol = 1e-4,
atol = my.atol, jacfunc = exampjac, jactype = "fullusr", hmax = Inf)
)
all.equal(as.data.frame(out), as.data.frame(outJ)) # TRUE
diagnostics(out)
diagnostics(outJ) # shows what lsoda did internally
# }
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